(untitled) (Journal of Computational Mathematics, 2024)
In this paper, we develop and analyze a finite difference method for linear second-order stochastic boundary-value problems (SBVPs) driven by additive white noises. First we regularize the noise by the Wong-Zakai approximation and introduce a sequence of linear second-order SBVPs. We prove that the solution of the SBVP with regularized noise converges to the solution of the original SBVP with convergence order O(h) in the mean-square sense. To obtain a numerical solution, we apply the finite difference method to the stochastic BVP whose noise is piecewise constant approximation of the original noise. The approximate SBVP with regularized noise is shown to have better regularity than the original problem, which facilitates the convergence proof for the proposed scheme. Convergence analysis is presented based on the standard finite difference method for deterministic problems. More specifically, we prove that the finite difference solution converges at O(h) in the mean-square sense, when the second-order accurate three-point formulas to approximate the first and second derivatives are used. Finally, we present several numerical examples to validate the efficiency and accuracy of the proposed scheme.